I am just getting back into the realm of equations again, and have a question, that maybe someone more into the depths of this could help me with.

I am trying to find a formula, into which I input desired Distance (s) and Time (t), resulting in a given Initial Velocity (u) in relation to a negative Acceleration (-a), so that after going the given Distance (s), the Final Velocity (v) = 0.

What equations I think should help me achieve this somehow:

s = 1/2 * a * t^2 + (u * t)

v = u + a*t

OR in this case:

s = 1/2 * -a * t^2 + (u * t)

v = u - a*t

I know that as an example, if I wanted to go the Distance (s)= 100m in Time (t) = 10s, I would use the following values:

1.)0 = 20 - 2*10 [v = u - a*t]

2.)100m = -1 * 10^2 + (20 * 10) [s = 1/2 -a * t^2 + (u * t)]

3.) Now I could just increase the Time (t) as I want in relation to changing either the Initial Velocity (u) or Acceleration(a); For example, if I wanted the Distance(s) = 1000m, I could change Time(t) = 100, and Acceleration(a) = 0.2 .

But I struggle converting this knowledge and the relativity of the values to each other into a formula. I would grealty appreciate if anyone could help me out here.

I don't know where you pulled those values of acceleration and initial velocity from in your example, but for that formula to work you will need to have only one variable (unless you plan on programming simultaneous equations into your program, but I've not tried that before so I can't help there sorry ). This means you can either look for acceleration, or initial velocity.

Acceleration can be calculated as (v - u) / t

Initial velocity can be calculated as v - at

However, you may find these equations of motion useful:

s = (1/2)(u + v)t

s = ut + (1/2)at^2

v^2 = u^2 + 2as

v = u + at

a = (v - u) / t

Edit: also, for unnaccelerated motion there is s = vt